On the validity of integral localized approximation for on-axis zeroth-order Mathieu beams

•Comparisons between the beam shape coefficients (BSCs) are obtained by using either an exact procedure or a localized procedure, in the case of an on-axis zeroth-order Mathieu beams (zMBs).•The presence of an extra-cosine term in the propagation factor restricts the range of validity of localized approximation to the evaluation of BSCs in the case of an on-axis zMBs.•Numerical comparisons confirm that the exact and localized values of BSCs agree reasonably well for the axicon angle up to 10°.

In a recent paper on radiation pressure forces exerted on a homogenous spherical particle by zeroth-order Mathieu beams (zMBs), the integral localized approximation(ILA) was used to calculate the beam shape coefficients (BSCs) encoding the shape of the beams. Unfortunately, this method is valid only for beams with a propagating factor exp ( ± ikz). In the case of non-diffracting beams the propagation factor is exp (± ikcos α z) which involves an extra-cosine term, with α being the axicon angle. Due to this term it has been demonstrated that localized approximations, including ILA, provide a satisfactory description of the intended beam only if the axicon angle is small enough. Zeroth-order Mathieu beams pertain to this type of beams. The present paper is therefore devoted to a comparison between BSCs calculated with an exact procedure and those calculated using ILA, in order to determine a range of validity of the approximate procedure. As a side result, we also establish exact closed-form expressions to the evaluation of BSCs of zMBs.